32] 
TO THE THEORY OF SPHERICAL COORDINATES. 
223 
which we see from the preceding formulae is the form of the relation between the 
systems p x and p. And suppose, as before, fi 1 , v 1 are the cosines of the distances 
of the new points of reference X 2 , Y 1} Z x . 
We can immediately determine the relations that must exist between these 
coefficients, in order that they may actually denote such a transformation. For this 
purpose write 
a , b , c = j (94), 
a', b', c '=/, 
a", b", c "=j". 
Then the distance between the point P and any other point P' is given by the formula 
IT(p, p', q) IT(p 1} pi, ©) 
cos 8 
V [W(p, p, q) IT (p', p', q)} V {IT (p u p ly ®) W (p u p[ t 0)} 
...(95), 
diere 
q), W (/,/, q), q), q)...(96). 
But we must evidently have 
cos 8 = 
W {p u pi, qO 
(97), 
V [W{p ly p 1 , q x ) W (pi, pi, qO} 
or the quantities ® must be proportional to the quantities q, i.e. 
W(j,j, q) : W(j',j\ q) : W(j",j", q) : W (/, j", q) : W(j",j, q) : W(j",j, q) 
= 1 : 1 : 1 : X 2 : /¿j : i>q ...(98). 
And in precisely the same manner, if instead of x, y, z, x x , y Y , z x , in the above 
formulae, we had had £, ij, % : y 1 , ^i, the result would have been 
W(j, j, q) : IT (/, /, q) : W(j", j", q) : W(/, j", q) : W(j", j, q) : W (/', j, q) 
= fl : h : C : f :q:l) ...(99). 
It is hardly necessary to remark, that throughout the preceding formulae an 
expression, such as W(p, p', q), is proportional to either of the quantities 
x£' + yr)' + zH or x'g + y'rj + z'£, 
and may be changed into one of these multiplied by an arbitrary constant, which 
may be always made to disappear by a corresponding change in another quantity of 
the same form : thus, for instance, 
IT (p, p\ q) + y'y + z"C, . 
W (p, p, q) x i; + y 7] + ^ l 
but these forms being unsymmetrical, it is better in general not to introduce them. 
All the preceding expressions simplify exceedingly, reducing themselves in fact to 
the ordinary formulae for the transformation of rectangular coordinates in Geometry of 
three dimensions, for the case where the triangle XYZ has its sides and angles right 
angles. As this presents no difficulty, I shall not enter upon it at present.